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The prominent Bernstein -- von Mises (BvM) result claims that the posterior distribution after centering by the efficient estimator and standardizing by the square root of the total Fisher information is nearly standard normal. In particular, the prior completely washes out from the asymptotic posterior distribution. This fact is fundamental and justifies the Bayes approach from the frequentist viewpoint. In the nonparametric setup the situation changes dramatically and the impact of prior becomes essential even for the contraction of the posterior; see [vdV2008], [Bo2011], [CaNi2013,CaNi2014] for different models like Gaussian regression or i.i.d. model in different weak topologies. This paper offers another non-asymptotic approach to studying the behavior of the posterior for a special but rather popular and useful class of statistical models and for Gaussian priors. First we derive tight finite sample bounds on posterior contraction in terms of the so called effective dimension of the parameter space. Our main results describe the accuracy of Gaussian approximation of the posterior. In particular, we show that restricting to the class of all centrally symmetric credible sets around pMLE allows to get Gaussian approximation up to order (n^{-1}). We also show that the posterior distribution mimics well the distribution of the penalized maximum likelihood estimator (pMLE) and reduce the question of reliability of credible sets to consistency of the pMLE-based confidence sets. The obtained results are specified for nonparametric log-density estimation and generalized regression.
In this paper, we study the asymptotic posterior distribution of linear functionals of the density. In particular, we give general conditions to obtain a semiparametric version of the Bernstein-Von Mises theorem. We then apply this general result to
We prove a Bernstein-von Mises theorem for a general class of high dimensional nonlinear Bayesian inverse problems in the vanishing noise limit. We propose a sufficient condition on the growth rate of the number of unknown parameters under which the
Few methods in Bayesian non-parametric statistics/ machine learning have received as much attention as Bayesian Additive Regression Trees (BART). While BART is now routinely performed for prediction tasks, its theoretical properties began to be under
Deheuvels [J. Multivariate Anal. 11 (1981) 102--113] and Genest and R{e}millard [Test 13 (2004) 335--369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cram{e}r--von Mises
The saddlepoint approximation gives an approximation to the density of a random variable in terms of its moment generating function. When the underlying random variable is itself the sum of $n$ unobserved i.i.d. terms, the basic classical result is t